protes 0.2.0

Method PROTES (PRobability Optimizer with TEnsor Sampling) for optimization of the multidimensional arrays and discretized multivariable functions based on the tensor train (TT) format

PROTES

Description

Method PROTES (PRobability Optimizer with TEnsor Sampling) for optimization of the multidimensional arrays and discretized multivariable functions based on the tensor train (TT) format.

Installation

To use this package, please install manually first the python programming language of the version >= 3.6, the jax framework and the optax library. Then, the package can be installed via pip: pip install protes.

Usage

Let's say we want to find the minimum of a d = 10 dimensional array that has n = 5 elements in each mode. Let an arbitrary array element y related to the d-dimensional multi-index i is defined by the function y = f(i) = sum(i). In this case, the optimizer may be launched in the following way:

import jax.numpy as jnp
from protes import protes
f_batch = lambda I: jnp.sum(I, axis=1)
i_opt, y_opt = protes(f=f_batch, d=10, n=5, m=5.E+3, log=True)

The function f_batch takes a set of multi-indices I (jax or numpy array having a size samples x d) and returns a list of the corresponding array values; m is is the computational budget (the allowed number of requested array elements). Returned values i_opt and y_opt are the found multi-index and the value in this multi-index for the approximate optimum, respectively.

Note that the code runs orders of magnitude faster if the array's mode size (n) is constant. If you need to optimize an array with discriminating mod sizes, then you should use the slow protes_general method. In this case, instead of two parameters d and n, one parameter n should be passed, which is a list of length d corresponding to the mod sizes in each dimension.

The demo folder contains several examples of using the PROTES method for real tasks (a simple demo can be run in the console with a command python demo/demo_func.py). This folder also contains a script animation.py for building an animation of the 2D optimization process (numpy and matplotlib libraries should be installed to run it).

Parameters of the protes function

• f (function) - the target function f(I), where input I is a 2D array of the shape [samples, d] (d is a number of dimensions of the function's input and samples is a batch size of requested multi-indices). The function should return 1D array on the CPU or GPU of the length equals to samples (the values of the target function for all provided multi-indices).
• d (int) - number of array dimensions.
• n (int) - mod size for each array's dimension.
• m (int) - the number of allowed requests to the objective function.
• k (int) - the batch size for optimization (the default value is 50).
• k_top (int) - number of selected candidates for all batches (it should be < k; the default value is 5).
• k_gd (int) - number of gradient lifting iterations for each batch (the default value is 100).
• lr (float): learning rate for gradient lifting iterations (the default value is 1.E-4).
• r (int): TT-rank of the constructed probability TT-tensor (the default value is 5).
• seed (int): parameter for random generator (the default value is 42).
• is_max (bool): if flag is set, then maximization rather than minimization will be performed.
• log (bool): if flag is set, then the information about the progress of the algorithm will be printed after each improvement of the optimization result.
• log_ind (bool): if flag is set and log is True, then the current optimal multi-index will be printed every step.
• info (dict): optional dictionary, which will be filled with reference information about the process of the algorithm operation.
• P (list): optional initial probability tensor in the TT-format. If this parameter is not set, then a random initial TT-tensor will be generated. Note that this tensor will be changed inplace.

Notes

• You can use the outer cache for the values requested by the optimizer (that is, for each requested batch, check if any of the multi-indices have already been calculated), this can in some cases reduce the number of requests to the objective function.

• For a number of tasks, performance can be improved by switching to increased precision in the representation of numbers in jax; for this, at the beginning of the script, you should specify the code:

  from jax.config import config
  config.update('jax_enable_x64', True)
  

• If there is a GPU, the jax optimizer code will be automatically executed on it, however, in some cases it is more convenient to execute the optimizer code on the CPU, for this you should specify the following code at the beginning of the executable script:

  import os
  os.environ['JAX_PLATFORM_NAME'] = 'cpu'
  

Authors

• Anastasia Batsheva
• Andrei Chertkov
• Ivan Oseledets
• Gleb Ryzhakov

Citation

If you find our approach and/or code useful in your research, please consider citing:

@article{batsheva2023protes,
    author    = {Batsheva, Anastasia and Chertkov, Andrei  and Ryzhakov, Gleb and Oseledets, Ivan},
    year      = {2023},
    title     = {PROTES: Probabilistic Optimization with Tensor Sampling},
    journal   = {arXiv preprint arXiv:2301.12162},
    url       = {https://arxiv.org/pdf/2301.12162.pdf}
}

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